How much does CD player affect overall sound?

A wise person once told me that preamp, cables, amp or speakers will not PUT BACK what is not there from the start when asked about the importance of a great source.

A lot, but so does everything else. There really is something to system matching. That said, I would have to ask what you feel you are missing that a new CDP would bring to the table.

You’ve got some nice gear, however both the CJ and the B&W’s have a very unique house sound. I have found CJ preamps to be overly dark, and slow (IMHO based upon personal experience with the Premier 3 and the Premier 14). I’ve also found the B&W to be a tab dark for my liking, unless properly matched (based upon poorly executed dealer demo’s).

FWIW, I’d rather have a mediocre CDP with a great pre/amp/speaker combo, than the other way around (a great CDP with a mediocre pre/amp/speaker combo) any day.

Transnova said all and said it well.
Garbage in garbage out!

I usually try to select a component upon the needs of my system's needs first, and my preffs second.

As personal preffs go, I'll vote otherwise for the CJ & BW items the other poster said were 'dark' sounding to hem. I found nothing muddy or unclear with either items... CJPV14 or BW 804s. Not at all.

Perhaps if some other digital sources are in or about to be in the mix, a DAC might be in order to seek out. A great DAC will escalate the perfomance of of and improve upon, the current levels of the now CDP. Albeit a very good DAC IMO begins around $1K new, and goes up from there. So there's that to think about.

What does your rig need? What do you want yourself, sound wise? Well, then, there's your direction.

My audiophile friend of 30 years in the biz, both in audio and studio recording, summed that what you heard from your stereo was comprised of 70% speakers, 20% room and 10% components. This is probably pretty close. I might take a few % points from the speakers and give a little more to room and maybe a couple ticks more to components, but speakers by themselves certainly are at 50% or more.

Cambridge digital in general is pretty good. You may have to spend quite a bit to get better, or maybe even just different, but not necessarily better.